MA 105: Calculus Lecture 20 . Prof. B.V. Limaye IIT Bombay - - PowerPoint PPT Presentation

. .

MA 105: Calculus Lecture 20

• Prof. B.V. Limaye

IIT Bombay Friday, 17 March 2017

B.V. Limaye MA 105: Lec-20

. . Vector Algebra

We begin a part of this course known as ‘Vector Analysis’. It deals with scalar fields and vector fields. We begin with some concepts in ‘Vector Algebra’. In Lecture 12, we have defined the Euclidean space Rm := {x x x = (x1, . . . , xm) : xi ∈ R for i = 1, . . . , m}, where m ∈ N. An element of R1 := R is called a scalar, and an element of Rm is called a vector if m ≥ 2. Let x x x := (x1, . . . , xm), y y y := (y1, . . . , ym) ∈ Rm, and a ∈ R. We have already defined the sum x x x + y y y and the scalar multiple ax x

• x. Also, we have studied the scalar product of x

x x and y y y: x x x · y y y := x1y1 + · · · + xmym ∈ R. It is often called the dot product of x x x and y y y.

B.V. Limaye MA 105: Lec-20

. . Vector Product

Let m := 3, x x x := (x1, x2, x3) and y y y := (y1, y2, y3). We now define the vector product of x x x with y y y: x x x × y y y := (x2y3 − x3y2, x3y1 − x1y3, x1y2 − x2y1) ∈ R3. It is often called the cross product of x x x with y y

• y. Let

i i i := (1, 0, 0), j j j := (0, 1, 0), k k k := (0, 0, 1). Then x x x := x1i i i + x2j j j + x3k k k, y y y := y1i i i + y2j j j + y3k k k, and x x x × y y y =

• i

i i j j j k k k x1 x2 x3 y1 y2 y3

• .

Note: i i i × j j j = k k k, j j j × k k k = i i i, k k k × i i i = j j j.

B.V. Limaye MA 105: Lec-20

Properties of a determinant show that for every z z z ∈ R3, (x x x + y y y) × z z z = (x x x × z z z) + (y y y × z z z) and y y y × x x x = −(x x x × y y y). Let x x x ̸= 0 0 and y y y ̸= 0

• 0. It can be shown that

x x x × y y y = ∥x x x∥ ∥y y y∥(sin θ)n n n, where θ ∈ [0, π] is the angle between x x x and y y y, and n n n is the unit vector which is perpendicular to the plane containing x x x and y y y, and obeys the ‘right-hand rule’.

n x×y y x θ

Hence ∥x x x ×y y y∥ = the area of the parallelogram with sides x x x, y y y.

B.V. Limaye MA 105: Lec-20

. . Scalar Triple Product and Vector triple Product

Let x x x, y y y, z z z ∈ R3. Then x x x · (y y y × z z z) ∈ R and x x x × (y y y × z z z) ∈ R3 are called the scalar triple product and the vector triple product of x x x, y y y, z z z respectively. It is easy to see that if x x x = (x1, x2, x3), y y y = (y1, y2, y3), z z z = (z1, z2, z3), then x x x · (y y y × z z z) =

• x1

x2 x3 y1 y2 y3 z1 z2 z3

• .

Geometrically, x x x · (y y y × z z z) can be interpreted as the (signed) volume of the parallelopiped defined by the vectors x x x,y y y,z z z. One can prove the Lagrange formula x x x × (y y y × z z z) = (x x x · z z z)y y y − (x x x · y y y)z z z by considering each component of the LHS and the RHS.

B.V. Limaye MA 105: Lec-20

. . Scalar Fields and Vector Fields

A scalar field is an assignment of a scalar to each point in a region in the space. For example, the temperature at a point

• n the earth is a scalar field (defined on a subset of R3.)

A vector field is an assignment of a vector to each point in a region in the space. For example, the velocity field of a moving fluid is a vector field that associates a velocity vector to each point in the fluid. . Definition . . Let m ∈ N, and let D be a subset of Rm. A scalar field is a map from D to R. A vector field is a map from D to Rm. If m = 2, then it is called a vector field in the plane, and if m = 3, then it is called a vector field in the space.

B.V. Limaye MA 105: Lec-20

. . Smooth Scalar and Vector Fields

Suppose D is an open subset of Rm, that is, every point in D is an interior point of D. A scalar field f : D → R is called smooth if all first order partial derivatives of f exist and are continuous on D. The set

• f all smooth scalar fields on D is denoted by C 1(D). Similarly,

the set of all scalar fields on D having continuous partial derivatives of the first and second order is denoted by C 2(D). Let F F F : D → Rm be a vector field on D, and let F F F(x x x) = (F1(x x x), . . . , Fm(x x x)) for x x x ∈ D, where Fi : D → R is scalar field on D, i = 1, . . . , m. Then the vector field F F F is called smooth if each component scalar field Fi is smooth, that is, if each ∂Fi ∂xj , i, j = 1, . . . , m exists and is continuous on D.

B.V. Limaye MA 105: Lec-20

. . Gradient, Divergence and Curl

Let f be a smooth scalar field onD ⊂ R3. Then the vector field grad f := ∇f = (∂f ∂x , ∂f ∂y , ∂f ∂z ) defined on D is called the gradient field of f . Next, let F F F := (P, Q, R) be a smooth vector field on D ⊂ R3. The divergence field of F F F is the scalar field on D defined by divF F F := ∇ · F F F = ∂P ∂x + ∂Q ∂y + ∂R ∂z , and the curl field of F F F is the vector field on D defined by curlF F F := ∇ × F F F = (∂R ∂y − ∂Q ∂z , ∂P ∂z − ∂R ∂x , ∂Q ∂x − ∂P ∂y ) .

B.V. Limaye MA 105: Lec-20

As in the case of the cross product x x x × y y y, we may write ∇ × F F F = ∇ × (P, Q, R) =

• i

i i j j j k k k ∂ ∂x ∂ ∂y ∂ ∂z P Q R

• .

For any m ∈ N, we can define the gradient field grad f of a scalar field f on an open subset of Rm as well as the divergence field divF F F of a vector field F F F on an open subset of Rm in a similar manner. But the curl field curlF F F can only be defined for a vector field F F F on an open subset of R3. Of course, if D is an open subset of R and f is a smooth scalar field on D, then we can let F F F(x, y, z) := (f (x), 0, 0) for (x, y, z) ∈ D × R2, and define curl f := curlF F

• F. Also, if D is an
• pen subset of R2 and G

G G is a smooth vector field on D, then we can let F F F(x, y, z) := (G(x, y), 0) for (x, y, z) ∈ D × R, and define curlG G G := curlF F F.

B.V. Limaye MA 105: Lec-20

. . The GCD Sequence

Suppose the first and second order partial derivatives of f and

• f P, Q, R exist and are continuous on D. By the Mixed

Partials Theorem, we obtain (i) curl (grad f ) = ∇ × (∇f ) = 0 0: ( ∂2f ∂y∂z − ∂2f ∂z∂y , ∂2f ∂z∂x − ∂2f ∂x∂z , ∂2f ∂x∂y − ∂2f ∂y∂x ) = (0, 0, 0), and also (ii) div (curlF F F) = ∇ · (∇ × F F F) = 0: ∂ ∂x (∂R ∂y − ∂Q ∂z ) + ∂ ∂y (∂P ∂z − ∂R ∂x ) + ∂ ∂z (∂Q ∂x − ∂P ∂y ) = 0. Thus Gradient, Curl and Divergence give a sequence of maps {scalar fields }

grad

− → {vector fields }

curl

− → {vector fields }

div

− → {scalar fields } which satisfies curl (grad f ) = 0 0 and div (curlF F F) = 0, so that the successive composites are zero.

B.V. Limaye MA 105: Lec-20

The above phenomenon raises the following basic questions: (i) If G G G is a smooth vector field such that curlG G G = 0 0, then is G G G a gradient field, that is, is there a scalar field f such that G G G = grad f ? (ii) If H H H is smooth vector field such that divH H H = 0, then is H H H a curl field, that is, is there a vector field F F F such that H H H = curlF F F? These questions are reminiscent of the Fundamental theorem

• f Calculus, Part I, which answers the following question in the

affirmative: If g is a continuous function on [a, b], then is there an antiderivative of g, that is, is there a differentiable function f such that g = f ′? As such, the two questions raised above call for a suitable theory of integration, to which we now turn. Eventually, we shall come back to these questions.

B.V. Limaye MA 105: Lec-20

. . Laplacian

Let f be a smooth vector field on D ⊂ R3, and suppose the second order partials fxx, fyy, fzz exist on D. Let us consider the maps {scalar fields }

grad

− → {vector fields }

div

− → {scalar fields } . The Laplacian field of f is the scalar field defined on D by div (grad f ) := ∇ · (∇f ) = ∇2f = ∂2f ∂x2 + ∂2f ∂2y + ∂2f ∂2z . The Laplacian plays a very important role in the theory of partial differential equations, and its various applications.

B.V. Limaye MA 105: Lec-20

. . Paths

Let α, β ∈ R with α < β, and let m ∈ N. A path or a parametrized curve in Rm is a continuous map γ γ γ : [α, β] → Rm, that is, if γ γ γ = (γ1, . . . , γm), then γj : [α, β] → R is continuous for each j = 1, . . . , m. A path γ γ γ : [α, β] → Rm is called closed if γ γ γ(α) = γ γ γ(β). Such a path is also called a loop. A path γ γ γ : [α, β] → Rm is called simple if γ γ γ(t1) ̸= γ γ γ(t2) for all t1, t2 ∈ (α, β), t1 ̸= t2. If for t ∈ [α, β], dγ γ γ dt = γ γ γ′(t) := (γ′

1(t), . . . , γ′ m(t))

exists, then it is called the tangent vector to γ γ γ at t, and if it is nonzero, then ˆ t t t := γ γ γ′(t)/∥γ γ γ′(t)∥ is called the unit tangent vector to γ γ γ at t. (We write ˆ t t t instead of ˆ t t tγ

γ γ(t) for brevity.)

B.V. Limaye MA 105: Lec-20

Further, a path γ γ γ in Rm is called smooth or C 1 if each γj : [α, β] → R is continuously differentiable for j = 1, . . . , m; in case γ γ γ is a closed curve, that is, γ γ γ(α) = γ γ γ(β), we also require γ γ γ′(α) = γ γ γ′(β). A smooth path γ γ γ in Rm is called regular if γ γ γ′(t) ̸= 0 0 for all t ∈ [α, β], that is, if the unit tangent vector to γ γ γ exists at each t ∈ [α, β]. A path γ γ γ in Rm is called piecewise smooth if there are α := t0 < t1 < · · · < tn =: β such that γ γ γ is smooth on each [ti−1, ti], i = 1, . . . , n, and it is called piecewise regular, if γ γ γ is regular on each [ti−1, ti], i = 1, . . . , n. We shall assume hereafter that all paths are piecewise smooth, unless otherwise stated.

B.V. Limaye MA 105: Lec-20

. . Path-connected and Convex Subsets

A subset D of Rm is called path-connected if for every u u u,v v v ∈ D, there is a path γ γ γ : [α, β] → Rm such that γ γ γ(α) = u u u, γ γ γ(β) = v v v and γ γ γ(t) ∈ D for all t ∈ (α, β). In particular, a convex subset D of Rm is path-connected since for u u u,v v v ∈ D, there is the straight-line path γ : [0, 1] → Rm defined by γ γ γ(t) := u u u + t(v v v − u u u) ∈ D for t ∈ [0, 1]. Examples: (i) The subset {(x, y) ∈ R2 : x2 + y 2 ≤ 1} of R2 is path-connected; in fact it is convex. (ii) The subset {(x, y) ∈ R2 : 1/2 ≤ x2 + y 2 ≤ 1} of R2 is path-connected, but it is not convex. (iii) The subset {(x, y) ∈ R2 : x2 + y 2 ≤ 1} ∪ {(2, 0)} of R2 is not path-connected.

B.V. Limaye MA 105: Lec-20

Example Let a > 0. Define γ γ γ(t) := (a cos t, a sin t) for t ∈ [−π, π]. This path is called the standard parametrized circle in R2

• f radius a. Since γ

γ γ(−π) = γ γ γ(π) and γ γ γ′(t) := (−a sin t, a cos t) for t ∈ [−π, π], we see that γ γ γ is closed, simple, smooth, regular, and its unit tangent vector at t ∈ [−π, π] is ˆ t t t := (− sin t, cos t). If we let γ γ γ(t) := (a cos 2t, a sin 2t) for t ∈ [−π, π], then it is easy to see that γ γ γ is also closed, smooth, regular, and its unit tangent vector at t ∈ [−π, π] is (− sin 2t, cos 2t). Note that γ γ γ([−π, π]) = γ γ γ([−π, π]), that is, the functions γ γ γ and γ γ γ have the same range, namely {(x, y) ∈ R2 :x2 + y 2 =1}, although they are clearly different paths: one goes around the circle once, but the other does so twice.

B.V. Limaye MA 105: Lec-20